1. What are Function Machines?
A function machine takes an input and performs some mathematical operations to give an output. The operations may include the four mathematical operations: addition, subtraction, multiplication and division. In algebra, function machine is a way of writing rules using a flow diagram.
For example let’s take a function machine that takes an input number ‘m’ returns the output by adding 5 to the input. This function machine can be represented as shown below:
Some other examples of function machines:
The first step in solving the questions on function machine is to form equations using the flow diagram. If you have some unknown in your function machine, you can solve for the unknown using the equation given by the function machine.
1.1 Finding output using input
You need to form the equation in the forward direction (from left to right) using function machine and input to get the resulting output. Let us now work out this using the example below:
The above machine is taking an input as 2 and performing two mathematical operations to give an output. Let us now try to write the equation for the above machine starting left from the input and moving to right. Input 2 is multiplied by 2 and 6 is added to this result to get an output, so the equation will be:
2 × 2 + 6 = Output
Solving the above equation, you will get the output as 10.
You may also expect questions in 11 Plus exams where you might need to work out the input given the output in function machine, so let us now see how to form and solve the equation to get the input from given output.
1.2 Finding input using output
You need to form the equation in the reverse direction (from right to left) using function machine and output to get the input. Let us now work out this using the example below:
The above machine is taking input and performing two mathematical operations to give the output as 15. While writing the equation from right to left to work out input, you need to inverse the operations. So output 15 will be multiplied by 3 (inverse of division) and subtracted by 5 (inverse of addition) to get the input, and the equation will be:
15 × 3 - 5 = Input
Solving the above equation, you will get the input as 40.
1.3 Finding missing rules using input and output
Don’t get panicked if you’re given the input and output of the function machine and asked to work out the missing rules or operations in the machine. Write an equation (moving from left to right) using the input and output and try to solve for the missing rule.
Look at the example below:
The above machine is taking input as 25 and performing two mathematical operations to give the output as 900. To work out the value of x, write the equation (starting from left), input 25 is added to 5 and then multiplied by x to give output as 900, so the equation will be:
(25 + 5) × x = 900
Solving the above equation,
30x = 900
x = 900 ÷ 30
x = 30
So the function machine will be
